Recall that a certain type of object admits an ADE classification if there is a notion of equivalence relative to which equivalence classes of objects of the given type can be placed in one-to-one correspondence with the collection of simply laced Dynkin diagrams. The simply laced Dynkin diagrams have themselves been classified into two infinite families (denoted $A_n$ and $D_n$) and three exceptional examples (denoted E_6, E_7, and E_8). See the wikipedia page for more details.

There is a veritable laundry list of objects that admit an ADE classification. Examples include (modulo a number of qualifiers that I don't want to get into):

Semisimple Lie algebras

Conformal field theories

Tame quivers

Platonic solids

Positive definite quadratic forms on graphs

The list goes on. Accoding to the aforementioned wikipedia page, Vladmir Arnold asked in 1976 if there is a connection between these different kinds of objects which really explains why they all admit a common classification. The page also makes an offhand comment about how such a connection might be suggested by string theory.

I am hoping that somebody can explain some of the progress that has been made (if any) on Arnold's question. A good answer to this question is not one which explains the proofs that various different types of objects admit ADE classifications, nor one which aimlessly extends the list above. Rather, I would like to see someone take a collection of objects which on the surface are unrelated but which all have ADE classifications and then outline a deeper connection between them which at least suggests that they might all have a common classification. Bonus points if anyone can justify wikipedia's invocation of string theory in this context.

I could use some help with the tags. Is "combinatorics" appropriate?
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Paul SiegelJul 24 '10 at 22:07

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I have added the tag "ade-classifications" because if ever there was a topic because of its very ubiquity cannot easily be classified, it's this one!
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José Figueroa-O'FarrillJul 25 '10 at 0:22

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And while you're at it, also Kleinian surface singularities
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David LehaviJul 25 '10 at 5:33

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I always think the ADE classification becomes a little more conceptual when you think of it as classifying even definite lattices generated by roots, rather than Dynkin diagrams. It is easy to associate such a lattice to most objects appearing in your list (I will pass on CFTs, thank you :), and in the cases where there is a direct connection (e.g. singularities and Lie groups as in Brieskorn's paper) one can related those lattices naturally. I assume Arnold knew this :) so this isn't quite the answer to your question - but eg the wikipedia page makes it sound more mysterious than necessary.
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Arend BayerJul 25 '10 at 9:36

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Most of the connections I'm familiar with (Kleinian surface singularities, binary polyhedral groups, finite/tame quivers, finite Cohen-Macaulay type) come down to one of two related things: the existence of an (sub)additive function on a graph, or the equation $1/p+1/q+1/r \geq 1$. Idun Reiten's article in the Notices from 10 years ago or so makes these things explicit for quivers and quadratic forms.
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Graham LeuschkeJul 25 '10 at 11:28

The scene is given by compactifications of string theory on Kleinian orbifolds $M_\Gamma=\mathbb{C}^2/\Gamma$ where $\Gamma$ is a discrete subgroup of $SU(2)$. The space $M_\Gamma$ admits a Kleinian singularity at the origin. After studying this physical system, one is less surprised to see that Kleinian singularities, quiver diagrams, ALE spaces, ADE diagrams and 2 dimensional Conformal Field Theories all admit the same ADE classifications since they provide different descriptions of the same underlying physical system.

Michael Douglas and Gregory Moore have studied the compactification of string theory on Kleinian orbifold $M_\Gamma$ using D-branes as probes of the geometry.
D-branes are extended objects on which strings can end.
D-branes provide a physical description of the geometry in terms of supersymmetric gauge theories. Such supersymmetric gauge theories are efficiently summarized by a quiver diagram with a very natural physical interpretation: the nodes correspond to D-branes with specific gauge groups on them and the links between the nodes are open strings ending on the branes.

The minimal energy configurations (the vacua) of these supersymmetric gauge theories are obtained finding the extrema of a potential whose construction is equivalent to the hyperkhäler quotient construction of Asymptotic Locally Euclidian Spaces (ALE spaces) first obtained by Kronheimer. ALE spaces are HyperKähler four dimensional real manifolds whose anti-self-dual metrics are asymptotic to a Kleinian orbifold $M_\Gamma=\mathbb{C}^4/ \Gamma$. Physically ALE spaces described gravitational instantons.
ALE spaces provide small resolutions of the Kleinian singularities where the singular point is replaced by a system of spheres whose intersection matrix is equivalent to the Cartan matrix of an ADE Dynkin diagram. One can also consider Yang-Mills instantons on such spaces. The gauge group associated with the Yang-Mills instantons is given by the type of ADE diagram obtained by the resolution of the singularity. This was analyzed in the math literature by Kronheimer and Nakajima. Physically the ALE instantons moduli space is equivalent to the vacua of the gauge theory description of D-branes located at the singularities.

The link between D-branes on ALE spaces (or equivalently Kleinian singularities) and the ADE classification of two dimensional Conformal Field Theories (CFT) was studied by Lershe, Lutken and Schweigert. Although the geometry is singular, the CFT description is smooth. The 2 dimensional CFT is coming directly from the string description: as a string evolves it described a 2 dimensional surface called the string worldsheet. D-branes enter the CFT as boundary states. In the description of the CFT, one recovers Arnold's ADE list of simple isolated singularities.

Updates

I would like to comment on the non-stringy part of the question. This is motivated by the comments of Victor Protsak.

If one removes all the string theory interpretation in the discussion above. What is left is
Kronheimer's description of ALE spaces. Kronheimer's construction provides a beautiful realization of McKay's correspondence between Kleinian singularities, their crepant resolutions and ADE diagrams. This is reviewed in chapter 7 of Dominic Joyce's book "Compact Manifolds with Special Holonomy". From that perspective, the string theory description provides a physical interpretation of Kronheimer's construction and adds a natural link with quiver diagrams and 2 dimensional Conformal Field Theories.

This is mostly a terminological/context question: do you view Donaldson invariants and instantons on ALE spaces (as in the original work of Donaldson and Kronheimer) to be part of string theory?
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Victor ProtsakAug 6 '10 at 0:37

The word "instanton" was invented by Gerard T'Hooft (a physicist) and the first examples of instantons on ALE were obtained by physicists as noticed in Kronheimer's original paper on ALE. Donaldson original paper is called "Application of gauge theory to 4 dimensional topology". However, I think that Donaldson invariants and instantons on ALE are part of mathematics in their own right. They get specially handy in string theory where they correspond to concrete physical concepts.
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JMEAug 6 '10 at 1:49

The reason I asked is because to me, at least, they are part of gauge theory, so I am not sure which part of your answer truly requires $D$-branes.
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Victor ProtsakAug 6 '10 at 5:34

Without D-branes there is no story at all connected the Kleinian singularities to the instanton moduli space. The tamed quiver is also a direct description of the D-brane picture. Finally, without D-branes there is also no reason to connect the Kleinian singularity to a 2 dimensional CFT. Why on earth a singularr 10 dimensional system will have a smooth 2 dimensional CFT description? It is an impressive aspect of the string description that the singular geometry is described by a smooth CFT. I have edited the text to make all these points clear.
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JMEAug 6 '10 at 11:11

This is great - thanks! If I wanted to read more, would you recommend going directly to Douglas, Moore, Kronheimer, and Nakajima? Or are there better references?
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Paul SiegelAug 7 '10 at 22:58